In general, if a set of three hermitian operators $J_i (i=1,2,3)$ satisfy the commutation relation $[J_i, J_j]=i\hbar\epsilon_{ijk}J_k$, then it follows that the eigenvalues of $j(j+1)\hbar^2$ of $J^2\equiv J_1^2+J_2^2+J_3^2$ are given by $j=0,\frac{1}{2},1,\frac{3}{2},2,...$
The set of orbital angular momentum $L_i=\epsilon_{ijk}x_j p_k$ satisfy this commutation relation. Therefore, the eigenvalues of $L^2$ is expected to carry both integral and half-integral eigenvalues in absence of any extra conditions. But to my knowledge, in every problem, $\ell$ takes only integral values $\ell=0,1,2,3...$
What causes $\ell$ not to have half-integral values? Can it be answered using a system-independent argument?
